The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion

The problem of deriving a gradient flow structure for the porous medium equation which is thermodynamic, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero-range process with jump rate g(k) = k(alpha), alpha > 1 is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key super-exponential estimate is obtained using pathwise discretised regularity estimates in the spirit of the Aubin-Lions-Simons lemma. This allows to exhibit the porous medium equation as the gradient flow of the entropy in a thermodynamic metric via the energy-dissipation inequality.